In the category of metrics with conical singularities along a smooth divisor with angle in ( 0 , 2 π ) , we show that locally defined weak solutions ( C 1 , 1 - solutions) to the Kähler–Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coordinates. This shows the weak Kähler–Einstein metrics constructed by Guenancia–Păun, and independently by Yao, are all actually strong-conical Kähler–Einstein metrics. The key step is to establish a Liouville-type theorem for weak-conical Kähler–Ricci flat metrics defined over ℂ n , which depends on a Calderon–Zygmund theory in the conical setting. The regularity of globally defined weak-conical Kähler–Einstein metrics is already proved by Guenancia–Paun using a different method.